This is really 3 problems. I think this is the easiest order to do them in:

A. derive (m3) in system S5
B. derive (m4) in system S5
C. derive (m5) in the combination of systems M and Brouwer.

A and B will prove that from S5 you get Brouwer & M combined; and C will prove that from Brouwer & M combined you can get S5.

For A: note that we've proved lots of times that (P → <>P). Consider your available axioms and chain rule.

For B: this one is hard. But I don't want to make it too easy, so here's a half hint. First, from the S5 dual and necessitation and some M2 (previously called M1), you can get:

• ([]<>[]P → [][]P)

Now note, we have proven that in S5 we have as a theorem axiom m3. What instance of this theorem might be helpful to apply chain rule to?

For C: OK, this one is less hard, but still challenging. Some tricks will be required! Consider this. The following is a theorem we can prove using M4: (<><>P --> <>P). Since it's a theorem, you have by necessitation that [](<><>P --> <>P) (for justification you can write, "theorem, necessitation"). Now, consider what you'll get using axiom M2 if that is the antecedent of an instance of M2. With a funny instance of M3, you can do chain rule for the finish.